Some properties of averaging simulated optimization methods 235
and thus
αμω μμ βλ
γ λ μ
ˆ ˆ ˆˆˆˆ
ˆ(^1) ΩΩ^11 ˆˆ
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟ i
i.e.,
α
λ
β
γ
α
β
γ
1
ˆ
ˆ
ˆ
ˆ
ˆ
h ˆ
Also,
σωω
σ
λ γ
2
2
2
11
ˆˆˆ
ˆ ˆ
Ω
h
Thus, we notice immediately that the active return α and the TE
2
are given by our
earlier results plus an additional term. Under the normality assumption, we
again examine some of the statistical properties of these new estimations.
We present the results below; the proofs are straightforward extensions of our
earlier results.
EEE
E
Var Var Var
() () ( /)
() ( / )
() () ( /
ααβγ
α βγ
ααβγ
ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆˆˆ), ˆ/ˆ
var(ˆ)((ˆ ))
since a andβγare independent
α
γ
1
(^11)
T
Eh
E(() (σσ ˆ )
γ
(^22) E TN^1
T
10.5 Remark 2
The portfolio optimization in Equation (10.11) above is the same as that consid-
ered in Okhrin and Schmid (2006). The expression for E()α can be derived from
the results in their appendix. In particular, using the expression for E()ωμˆEU|,ˆ
given on page 248 of their paper, multiplying by μˆ and then taking expecta-
tions, we will get the same result as detailed above once we allow for the differ-
ence in the definition of ωˆ 1
0 ; their divisor is T 1 rather than T.